\(\int \frac {a+b \arcsin (c x)}{x (d+e x^2)} \, dx\) [455]

Optimal result

Integrand size = 21, antiderivative size = 518 \[ \int \frac {a+b \arcsin (c x)}{x \left (d+e x^2\right )} \, dx=-\frac {(a+b \arcsin (c x)) \log \left (1-\frac {\sqrt {e} e^{i \arcsin (c x)}}{i c \sqrt {-d}-\sqrt {c^2 d+e}}\right )}{2 d}-\frac {(a+b \arcsin (c x)) \log \left (1+\frac {\sqrt {e} e^{i \arcsin (c x)}}{i c \sqrt {-d}-\sqrt {c^2 d+e}}\right )}{2 d}-\frac {(a+b \arcsin (c x)) \log \left (1-\frac {\sqrt {e} e^{i \arcsin (c x)}}{i c \sqrt {-d}+\sqrt {c^2 d+e}}\right )}{2 d}-\frac {(a+b \arcsin (c x)) \log \left (1+\frac {\sqrt {e} e^{i \arcsin (c x)}}{i c \sqrt {-d}+\sqrt {c^2 d+e}}\right )}{2 d}+\frac {(a+b \arcsin (c x)) \log \left (1-e^{2 i \arcsin (c x)}\right )}{d}+\frac {i b \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{i \arcsin (c x)}}{i c \sqrt {-d}-\sqrt {c^2 d+e}}\right )}{2 d}+\frac {i b \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{i \arcsin (c x)}}{i c \sqrt {-d}-\sqrt {c^2 d+e}}\right )}{2 d}+\frac {i b \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{i \arcsin (c x)}}{i c \sqrt {-d}+\sqrt {c^2 d+e}}\right )}{2 d}+\frac {i b \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{i \arcsin (c x)}}{i c \sqrt {-d}+\sqrt {c^2 d+e}}\right )}{2 d}-\frac {i b \operatorname {PolyLog}\left (2,e^{2 i \arcsin (c x)}\right )}{2 d} \] Output:

-1/2*(a+b*arcsin(c*x))*ln(1-e^(1/2)*(I*c*x+(-c^2*x^2+1)^(1/2))/(I*c*(-d)^( 
1/2)-(c^2*d+e)^(1/2)))/d-1/2*(a+b*arcsin(c*x))*ln(1+e^(1/2)*(I*c*x+(-c^2*x 
^2+1)^(1/2))/(I*c*(-d)^(1/2)-(c^2*d+e)^(1/2)))/d-1/2*(a+b*arcsin(c*x))*ln( 
1-e^(1/2)*(I*c*x+(-c^2*x^2+1)^(1/2))/(I*c*(-d)^(1/2)+(c^2*d+e)^(1/2)))/d-1 
/2*(a+b*arcsin(c*x))*ln(1+e^(1/2)*(I*c*x+(-c^2*x^2+1)^(1/2))/(I*c*(-d)^(1/ 
2)+(c^2*d+e)^(1/2)))/d+(a+b*arcsin(c*x))*ln(1-(I*c*x+(-c^2*x^2+1)^(1/2))^2 
)/d+1/2*I*b*polylog(2,-e^(1/2)*(I*c*x+(-c^2*x^2+1)^(1/2))/(I*c*(-d)^(1/2)- 
(c^2*d+e)^(1/2)))/d+1/2*I*b*polylog(2,e^(1/2)*(I*c*x+(-c^2*x^2+1)^(1/2))/( 
I*c*(-d)^(1/2)-(c^2*d+e)^(1/2)))/d+1/2*I*b*polylog(2,-e^(1/2)*(I*c*x+(-c^2 
*x^2+1)^(1/2))/(I*c*(-d)^(1/2)+(c^2*d+e)^(1/2)))/d+1/2*I*b*polylog(2,e^(1/ 
2)*(I*c*x+(-c^2*x^2+1)^(1/2))/(I*c*(-d)^(1/2)+(c^2*d+e)^(1/2)))/d-1/2*I*b* 
polylog(2,(I*c*x+(-c^2*x^2+1)^(1/2))^2)/d
 

Mathematica [A] (verified)

Time = 0.14 (sec) , antiderivative size = 430, normalized size of antiderivative = 0.83 \[ \int \frac {a+b \arcsin (c x)}{x \left (d+e x^2\right )} \, dx=-\frac {b \arcsin (c x) \log \left (1+\frac {\sqrt {e} e^{i \arcsin (c x)}}{c \sqrt {d}-\sqrt {c^2 d+e}}\right )+b \arcsin (c x) \log \left (1+\frac {\sqrt {e} e^{i \arcsin (c x)}}{-c \sqrt {d}+\sqrt {c^2 d+e}}\right )+b \arcsin (c x) \log \left (1-\frac {\sqrt {e} e^{i \arcsin (c x)}}{c \sqrt {d}+\sqrt {c^2 d+e}}\right )+b \arcsin (c x) \log \left (1+\frac {\sqrt {e} e^{i \arcsin (c x)}}{c \sqrt {d}+\sqrt {c^2 d+e}}\right )-2 b \arcsin (c x) \log \left (1-e^{2 i \arcsin (c x)}\right )-2 a \log (x)+a \log \left (d+e x^2\right )-i b \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{i \arcsin (c x)}}{c \sqrt {d}-\sqrt {c^2 d+e}}\right )-i b \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{i \arcsin (c x)}}{-c \sqrt {d}+\sqrt {c^2 d+e}}\right )-i b \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{i \arcsin (c x)}}{c \sqrt {d}+\sqrt {c^2 d+e}}\right )-i b \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{i \arcsin (c x)}}{c \sqrt {d}+\sqrt {c^2 d+e}}\right )+i b \operatorname {PolyLog}\left (2,e^{2 i \arcsin (c x)}\right )}{2 d} \] Input:

Integrate[(a + b*ArcSin[c*x])/(x*(d + e*x^2)),x]
 

Output:

-1/2*(b*ArcSin[c*x]*Log[1 + (Sqrt[e]*E^(I*ArcSin[c*x]))/(c*Sqrt[d] - Sqrt[ 
c^2*d + e])] + b*ArcSin[c*x]*Log[1 + (Sqrt[e]*E^(I*ArcSin[c*x]))/(-(c*Sqrt 
[d]) + Sqrt[c^2*d + e])] + b*ArcSin[c*x]*Log[1 - (Sqrt[e]*E^(I*ArcSin[c*x] 
))/(c*Sqrt[d] + Sqrt[c^2*d + e])] + b*ArcSin[c*x]*Log[1 + (Sqrt[e]*E^(I*Ar 
cSin[c*x]))/(c*Sqrt[d] + Sqrt[c^2*d + e])] - 2*b*ArcSin[c*x]*Log[1 - E^((2 
*I)*ArcSin[c*x])] - 2*a*Log[x] + a*Log[d + e*x^2] - I*b*PolyLog[2, (Sqrt[e 
]*E^(I*ArcSin[c*x]))/(c*Sqrt[d] - Sqrt[c^2*d + e])] - I*b*PolyLog[2, (Sqrt 
[e]*E^(I*ArcSin[c*x]))/(-(c*Sqrt[d]) + Sqrt[c^2*d + e])] - I*b*PolyLog[2, 
-((Sqrt[e]*E^(I*ArcSin[c*x]))/(c*Sqrt[d] + Sqrt[c^2*d + e]))] - I*b*PolyLo 
g[2, (Sqrt[e]*E^(I*ArcSin[c*x]))/(c*Sqrt[d] + Sqrt[c^2*d + e])] + I*b*Poly 
Log[2, E^((2*I)*ArcSin[c*x])])/d
 

Rubi [A] (verified)

Time = 1.69 (sec) , antiderivative size = 518, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {5232, 2009}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \frac {a+b \arcsin (c x)}{x \left (d+e x^2\right )} \, dx\)

\(\Big \downarrow \) 5232

\(\displaystyle \int \left (\frac {a+b \arcsin (c x)}{d x}-\frac {e x (a+b \arcsin (c x))}{d \left (d+e x^2\right )}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {(a+b \arcsin (c x)) \log \left (1-\frac {\sqrt {e} e^{i \arcsin (c x)}}{-\sqrt {c^2 d+e}+i c \sqrt {-d}}\right )}{2 d}-\frac {(a+b \arcsin (c x)) \log \left (1+\frac {\sqrt {e} e^{i \arcsin (c x)}}{-\sqrt {c^2 d+e}+i c \sqrt {-d}}\right )}{2 d}-\frac {(a+b \arcsin (c x)) \log \left (1-\frac {\sqrt {e} e^{i \arcsin (c x)}}{\sqrt {c^2 d+e}+i c \sqrt {-d}}\right )}{2 d}-\frac {(a+b \arcsin (c x)) \log \left (1+\frac {\sqrt {e} e^{i \arcsin (c x)}}{\sqrt {c^2 d+e}+i c \sqrt {-d}}\right )}{2 d}+\frac {\log \left (1-e^{2 i \arcsin (c x)}\right ) (a+b \arcsin (c x))}{d}+\frac {i b \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{i \arcsin (c x)}}{i c \sqrt {-d}-\sqrt {d c^2+e}}\right )}{2 d}+\frac {i b \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{i \arcsin (c x)}}{i c \sqrt {-d}-\sqrt {d c^2+e}}\right )}{2 d}+\frac {i b \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{i \arcsin (c x)}}{i \sqrt {-d} c+\sqrt {d c^2+e}}\right )}{2 d}+\frac {i b \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{i \arcsin (c x)}}{i \sqrt {-d} c+\sqrt {d c^2+e}}\right )}{2 d}-\frac {i b \operatorname {PolyLog}\left (2,e^{2 i \arcsin (c x)}\right )}{2 d}\)

Input:

Int[(a + b*ArcSin[c*x])/(x*(d + e*x^2)),x]
 

Output:

-1/2*((a + b*ArcSin[c*x])*Log[1 - (Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d 
] - Sqrt[c^2*d + e])])/d - ((a + b*ArcSin[c*x])*Log[1 + (Sqrt[e]*E^(I*ArcS 
in[c*x]))/(I*c*Sqrt[-d] - Sqrt[c^2*d + e])])/(2*d) - ((a + b*ArcSin[c*x])* 
Log[1 - (Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] + Sqrt[c^2*d + e])])/(2* 
d) - ((a + b*ArcSin[c*x])*Log[1 + (Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d 
] + Sqrt[c^2*d + e])])/(2*d) + ((a + b*ArcSin[c*x])*Log[1 - E^((2*I)*ArcSi 
n[c*x])])/d + ((I/2)*b*PolyLog[2, -((Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[ 
-d] - Sqrt[c^2*d + e]))])/d + ((I/2)*b*PolyLog[2, (Sqrt[e]*E^(I*ArcSin[c*x 
]))/(I*c*Sqrt[-d] - Sqrt[c^2*d + e])])/d + ((I/2)*b*PolyLog[2, -((Sqrt[e]* 
E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] + Sqrt[c^2*d + e]))])/d + ((I/2)*b*PolyLo 
g[2, (Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] + Sqrt[c^2*d + e])])/d - (( 
I/2)*b*PolyLog[2, E^((2*I)*ArcSin[c*x])])/d
 

Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 5232

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_.)*((d_) + (e_ 
.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*ArcSin[c*x])^n, ( 
f*x)^m*(d + e*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[c^2*d + 
 e, 0] && IGtQ[n, 0] && IntegerQ[p] && IntegerQ[m]
 

Maple [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 2.33 (sec) , antiderivative size = 344, normalized size of antiderivative = 0.66

method	result	size
parts	\(\frac {a \ln \left (x \right )}{d}-\frac {a \ln \left (e \,x^{2}+d \right )}{2 d}+b \left (\frac {\arcsin \left (c x \right ) \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )}{d}-\frac {i \operatorname {dilog}\left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )}{d}+\frac {i \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (e \,\textit {\_Z}^{4}+\left (-4 c^{2} d -2 e \right ) \textit {\_Z}^{2}+e \right )}{\sum }\frac {\left (\textit {\_R1}^{2} e -4 c^{2} d -e \right ) \left (i \arcsin \left (c x \right ) \ln \left (\frac {\textit {\_R1} -i c x -\sqrt {-c^{2} x^{2}+1}}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {\textit {\_R1} -i c x -\sqrt {-c^{2} x^{2}+1}}{\textit {\_R1}}\right )\right )}{\textit {\_R1}^{2} e -2 c^{2} d -e}\right )}{4 d}+\frac {i \operatorname {dilog}\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )}{d}+\frac {i \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (e \,\textit {\_Z}^{4}+\left (-4 c^{2} d -2 e \right ) \textit {\_Z}^{2}+e \right )}{\sum }\frac {\left (\textit {\_R1}^{2}-1\right ) \left (i \arcsin \left (c x \right ) \ln \left (\frac {\textit {\_R1} -i c x -\sqrt {-c^{2} x^{2}+1}}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {\textit {\_R1} -i c x -\sqrt {-c^{2} x^{2}+1}}{\textit {\_R1}}\right )\right )}{\textit {\_R1}^{2} e -2 c^{2} d -e}\right ) e}{4 d}\right )\)	\(344\)
derivativedivides	\(\frac {a \ln \left (c x \right )}{d}-\frac {a \ln \left (c^{2} e \,x^{2}+c^{2} d \right )}{2 d}+\frac {i b \operatorname {dilog}\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )}{d}+\frac {b \arcsin \left (c x \right ) \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )}{d}-\frac {i b \operatorname {dilog}\left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )}{d}+\frac {i b \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (e \,\textit {\_Z}^{4}+\left (-4 c^{2} d -2 e \right ) \textit {\_Z}^{2}+e \right )}{\sum }\frac {\left (-\textit {\_R1}^{2} e +4 c^{2} d +e \right ) \left (i \arcsin \left (c x \right ) \ln \left (\frac {\textit {\_R1} -i c x -\sqrt {-c^{2} x^{2}+1}}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {\textit {\_R1} -i c x -\sqrt {-c^{2} x^{2}+1}}{\textit {\_R1}}\right )\right )}{-\textit {\_R1}^{2} e +2 c^{2} d +e}\right )}{4 d}-\frac {i b \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (e \,\textit {\_Z}^{4}+\left (-4 c^{2} d -2 e \right ) \textit {\_Z}^{2}+e \right )}{\sum }\frac {\left (\textit {\_R1}^{2}-1\right ) \left (i \arcsin \left (c x \right ) \ln \left (\frac {\textit {\_R1} -i c x -\sqrt {-c^{2} x^{2}+1}}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {\textit {\_R1} -i c x -\sqrt {-c^{2} x^{2}+1}}{\textit {\_R1}}\right )\right )}{-\textit {\_R1}^{2} e +2 c^{2} d +e}\right ) e}{4 d}\)	\(352\)
default	\(\frac {a \ln \left (c x \right )}{d}-\frac {a \ln \left (c^{2} e \,x^{2}+c^{2} d \right )}{2 d}+\frac {i b \operatorname {dilog}\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )}{d}+\frac {b \arcsin \left (c x \right ) \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )}{d}-\frac {i b \operatorname {dilog}\left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )}{d}+\frac {i b \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (e \,\textit {\_Z}^{4}+\left (-4 c^{2} d -2 e \right ) \textit {\_Z}^{2}+e \right )}{\sum }\frac {\left (-\textit {\_R1}^{2} e +4 c^{2} d +e \right ) \left (i \arcsin \left (c x \right ) \ln \left (\frac {\textit {\_R1} -i c x -\sqrt {-c^{2} x^{2}+1}}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {\textit {\_R1} -i c x -\sqrt {-c^{2} x^{2}+1}}{\textit {\_R1}}\right )\right )}{-\textit {\_R1}^{2} e +2 c^{2} d +e}\right )}{4 d}-\frac {i b \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (e \,\textit {\_Z}^{4}+\left (-4 c^{2} d -2 e \right ) \textit {\_Z}^{2}+e \right )}{\sum }\frac {\left (\textit {\_R1}^{2}-1\right ) \left (i \arcsin \left (c x \right ) \ln \left (\frac {\textit {\_R1} -i c x -\sqrt {-c^{2} x^{2}+1}}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {\textit {\_R1} -i c x -\sqrt {-c^{2} x^{2}+1}}{\textit {\_R1}}\right )\right )}{-\textit {\_R1}^{2} e +2 c^{2} d +e}\right ) e}{4 d}\)	\(352\)

Input:

int((a+b*arcsin(c*x))/x/(e*x^2+d),x,method=_RETURNVERBOSE)
 

Output:

a/d*ln(x)-1/2*a/d*ln(e*x^2+d)+b*(1/d*arcsin(c*x)*ln(1+I*c*x+(-c^2*x^2+1)^( 
1/2))-I/d*dilog(1+I*c*x+(-c^2*x^2+1)^(1/2))+1/4*I*sum((_R1^2*e-4*c^2*d-e)/ 
(_R1^2*e-2*c^2*d-e)*(I*arcsin(c*x)*ln((_R1-I*c*x-(-c^2*x^2+1)^(1/2))/_R1)+ 
dilog((_R1-I*c*x-(-c^2*x^2+1)^(1/2))/_R1)),_R1=RootOf(e*_Z^4+(-4*c^2*d-2*e 
)*_Z^2+e))/d+I*dilog(I*c*x+(-c^2*x^2+1)^(1/2))/d+1/4*I*sum((_R1^2-1)/(_R1^ 
2*e-2*c^2*d-e)*(I*arcsin(c*x)*ln((_R1-I*c*x-(-c^2*x^2+1)^(1/2))/_R1)+dilog 
((_R1-I*c*x-(-c^2*x^2+1)^(1/2))/_R1)),_R1=RootOf(e*_Z^4+(-4*c^2*d-2*e)*_Z^ 
2+e))*e/d)
 

Fricas [F]

\[ \int \frac {a+b \arcsin (c x)}{x \left (d+e x^2\right )} \, dx=\int { \frac {b \arcsin \left (c x\right ) + a}{{\left (e x^{2} + d\right )} x} \,d x } \] Input:

integrate((a+b*arcsin(c*x))/x/(e*x^2+d),x, algorithm="fricas")
 

Output:

integral((b*arcsin(c*x) + a)/(e*x^3 + d*x), x)
 

Sympy [F]

\[ \int \frac {a+b \arcsin (c x)}{x \left (d+e x^2\right )} \, dx=\int \frac {a + b \operatorname {asin}{\left (c x \right )}}{x \left (d + e x^{2}\right )}\, dx \] Input:

integrate((a+b*asin(c*x))/x/(e*x**2+d),x)
 

Output:

Integral((a + b*asin(c*x))/(x*(d + e*x**2)), x)
 

Maxima [F]

\[ \int \frac {a+b \arcsin (c x)}{x \left (d+e x^2\right )} \, dx=\int { \frac {b \arcsin \left (c x\right ) + a}{{\left (e x^{2} + d\right )} x} \,d x } \] Input:

integrate((a+b*arcsin(c*x))/x/(e*x^2+d),x, algorithm="maxima")
 

Output:

-1/2*a*(log(e*x^2 + d)/d - 2*log(x)/d) + b*integrate(arctan2(c*x, sqrt(c*x 
 + 1)*sqrt(-c*x + 1))/(e*x^3 + d*x), x)
 

Giac [F(-2)]

Exception generated. \[ \int \frac {a+b \arcsin (c x)}{x \left (d+e x^2\right )} \, dx=\text {Exception raised: RuntimeError} \] Input:

integrate((a+b*arcsin(c*x))/x/(e*x^2+d),x, algorithm="giac")
 

Output:

Exception raised: RuntimeError >> an error occurred running a Giac command 
:INPUT:sage2OUTPUT:sym2poly/r2sym(const gen & e,const index_m & i,const ve 
cteur & l) Error: Bad Argument Value
                                                                                    
                                                                                    
 

Mupad [F(-1)]

Timed out. \[ \int \frac {a+b \arcsin (c x)}{x \left (d+e x^2\right )} \, dx=\int \frac {a+b\,\mathrm {asin}\left (c\,x\right )}{x\,\left (e\,x^2+d\right )} \,d x \] Input:

int((a + b*asin(c*x))/(x*(d + e*x^2)),x)
 

Output:

int((a + b*asin(c*x))/(x*(d + e*x^2)), x)
 

Reduce [F]

\[ \int \frac {a+b \arcsin (c x)}{x \left (d+e x^2\right )} \, dx=\frac {2 \left (\int \frac {\mathit {asin} \left (c x \right )}{e \,x^{3}+d x}d x \right ) b d -\mathrm {log}\left (e \,x^{2}+d \right ) a +2 \,\mathrm {log}\left (x \right ) a}{2 d} \] Input:

int((a+b*asin(c*x))/x/(e*x^2+d),x)
 

Output:

(2*int(asin(c*x)/(d*x + e*x**3),x)*b*d - log(d + e*x**2)*a + 2*log(x)*a)/( 
2*d)